Chern-Simons invariants, SO(3) instantons, and Z/2-homology cobordism

Transcription

1 Chern-Simons invariants, SO(3) instantons, and Z/2-homology cobordism Matthew Hedden and Paul Kirk Abstract. We review the SO(3) instanton gauge theory of Fintushel and Stern and recast it in the context of 4-manifolds with cylindrical ends. Applications to the Z/2 homology cobordism group of Z/2 homology 3-spheres are given. Contents 1. Introduction 1 2. Adapted bundles and instantons Adapted bundles The Pontryagin charge, instantons, and Chern-Simons invariants The moduli space Calculation of the index Compactness Reducible connections Enumeration of reducible instantons The argument of Furuta Computing Atiyah-Patodi-Singer ρ invariants and Chern-Simons invariants of flat SO(2) connections on rational homology spheres Seifert fibered examples Example: non Seifert fibered homology 3-spheres 29 References Introduction Among the many results of instanton gauge theory, one, due to Furuta, stands out because no alternative proof has been found despite enormous progress in gauge theory made through the Seiberg-Witten and Ozsváth-Szabó theories Mathematics Subject Classification. Primary 57M25. This work was supported in part by the National Science Foundation under grants , , , and

2 2 MATTHEW HEDDEN AND PAUL KIRK Theorem (Furuta [16], see also [14]). The Seifert fibered integer homology 3- spheres Σ(p, q, pqk 1), k = 1, 2, are linearly independent in the homology cobordism group, Θ 3 H. Furuta proved this theorem using the machinery of instantons on pseudofree orbifolds, as developed by Fintushel and Stern [13, 14]. In light of the (perhaps imprecise) expectation that the geometric information contained in Donaldson theory should coincide with that of the more recent approaches to gauge theory, the difficulty in proving this theorem in the context of Seiberg-Witten or Ozsváth-Szabó theory is quite mysterious, and motivates the present work. The aim of this article is to revisit the technique used in the proof of Furuta s result and to recast it in the light of advances in the theory of instantons on manifolds with cylindrical ends. It is our hope that understanding the result in this context will increase its power as a tool for studying smooth cobordism or knot concordance, and will perhaps shed light on how one could extract similar information from the modern invariants. Before describing some of the nuances involved in the cylindrical end reformulation, we present the following application afforded by the present approach (see Theorem 3.3 for a stronger statement and Section 4 for further applications). Theorem 1. Suppose that p, q, d are relatively prime positive odd integers. For k d large enough, the rational homology spheres obtained by d k 1 surgeries on the righthanded (p, q) torus knot are linearly independent in the Z/2 homology cobordism group Θ 3 Z/2. As mentioned, the mechanism underlying Furuta s argument relied on the use of orbifold connections and the machinery of instantons on manifolds with cylindrical ends, introduced in Taubes s ground-breaking article [33]. Here, we reformulate the entire argument in terms of gauge theory on 4-manifolds with cylindrical ends in the manner pioneered by Taubes [33] and Floer [15]. This yields additional flexibility, in particular providing cobordism obstructions outside of the realm of pseudofree orbifolds. In addition, however, our treatment allows us to carefully address several technical points. In the present context we are forced to consider the L 2 moduli space of instantons when addressing the issue of compactness, and hence must appeal to the results of [28]. Here, the Chern-Simons invariants of flat connections provide a lower bound on the quanta of energy that can escape in a non-convergent sequence of instantons. It is perhaps implicit (if not explicitly stated) in [16] and [14] that the boundary 3-manifolds should not admit degenerate flat SO(3) connections, and in any case this is automatic when the boundary manifolds are lens spaces or Seifert fibered homology spheres Σ(p, q, r) with three singular fibers. But for more general 3- manifolds extra care with hypotheses is needed, and we hope our exposition provides this. In addition, new wrinkles arise in the enumeration of reducible instantons (see Theorem 2.16). These and other points are presented in a context that eschews orbifolds in favor of manifolds with cylindrical ends. This material can also be considered as a generalization of the results of Ruberman and Matić, [27, 32], which correspond to the case p 1 (E, α) = 0. In a companion article [18], we will use this machinery to show that a certain infinite family of untwisted, positive-clasped Whitehead doubles of torus knots is

3 CHERN-SIMONS INVARIANTS, INSTANTONS, AND Z/2-HOMOLOGY COBORDISM 3 linearly independent in the smooth concordance group. We anticipate many more applications to the study of concordance and cobordism groups. Finally, we remark that the general argument is reminiscent of techniques that use Casson-Gordon invariants to establish linear independence in the topological knot concordance group [19], and this intriguing similarity provides motivation for further study. Roughly speaking, in the context of Casson-Gordon invariants arguments focus on one prime p at a time and corresponding D 2p (dihedral) representations. Knots whose branched covers have no p-torsion in their homology contribute nothing to the obstruction to linear independence. In the Furuta/Fintushel-Stern argument, one focuses on the smallest Chern-Simons invariant of flat SO(3) connections. Homology spheres whose smallest Chern-Simons invariant exceed this minimum contribute nothing to the obstruction to linear independence. Acknowledgments: It is our pleasure to thank Tom Mrowka, Charles Livingston, and Ron Fintushel for many interesting conversations. The second author thanks the organizers of the inspiring Chern-Simons Gauge Theory: 20 years after conference. 2. Adapted bundles and instantons 2.1. Adapted bundles. We begin by reviewing Donaldson s description [9] of gauge theory on adapted bundles over 4-manifolds with cylindrical ends, in the context of SO(3) bundles. Given a flat SO(3) connection α on a 3-manifold Y, denote by Hα (Y ) the cohomology of Y with coefficients in the corresponding flat R 3 bundle. The flat connection α is called non-degenerate if Hα 1 (Y ) = 0. If Y is a rational homology sphere, then the trivial connection is non-degenerate. Consider a compact oriented 4-manifold X with boundary X = Y = c i=1 Y i, a disjoint union of closed 3-manifolds. Endow X with a Riemannian metric which is isometric to [ 1, 0] Y in a collar neighborhood of the boundary. Throughout the rest of this section, we make the following assumption. The 4-manifold X is path connected, satisfies H 1 (X; Z/2) = 0, and has non-empty boundary Y = c i=1 Y i which is a disjoint union of rational homology 3-spheres. Form the non-compact manifold X = X Y ( [0, ) Y ) by adding an infinitely long collar to each boundary component. Choose a Riemannian metric on X whose restriction to each cylinder is the product metric. The submanifold [0, ) Y i will be called an end of X. Definition 2.1. An adapted bundle (E, α) over X is an SO(3) vector bundle E X, together with a fixed flat connection α i on each end [0, ) Y i. We use α as shorthand for the set {α i }. Two adapted bundles (E, α) and (E, α ) are called equivalent if there is a bundle isomorphism from E to E identifying the flat connections α and α over [r, ) Y for some r 0. Any adapted bundle (E, α) is equivalent to one in which the flat connection α i on each end [0, ) Y i is in cylindrical form, that is α i = π ( α i ) where α i is a flat connection on Y i and π : [0, ) Y i Y i is the projection to the second factor. Indeed, since α is flat, there always exists a gauge transformation g : E E which equals the identity on the interior of X so that g (α) has this form, obtained by parallel transport along rays [0, ) {y}. We will tacitly assume that each α i

4 4 MATTHEW HEDDEN AND PAUL KIRK is in cylindrical form whenever convenient, and use the same notation α i for the restriction of this flat connection to Y i. Since X has non-empty boundary, SO(3) vector bundles over X are isomorphic if and only if their second Stiefel-Whitney classes are equal. In particular, if (E, α) and (E, α ) are equivalent adapted bundles over X, then w 2 (E) = w 2 (E ) The Pontryagin charge, instantons, and Chern-Simons invariants. Given any SO(3) connection A on a bundle E over a (not necessarily compact) 4 manifold, Z, let F(A) denote its curvature 2-form. Define the Pontryagin charge of A to be the real number (2.1) p 1 (A) = 1 8π 2 Z Tr(F(A) F(A)), provided this integral converges. When Z is closed, p 1 (A) = p 1 (E), [Z] Z. Definition 2.2. An SO(3) connection A on a Riemannian manifold Z is called an instanton if its curvature form satisfies the equation If A is an instanton, then p 1 (A) = 1 8π 2 Z F(A) = F(A). Tr(F(A) F(A)) = 1 8π 2 F(A) 2 L 2 whence p 1 (A) is defined and non-negative for any instanton A whose curvature has finite L 2 norm. Given an adapted bundle (E, α) over X, choose an SO(3) connection A 0 on E which extends the given flat connection α on the union of the ends. Then its curvature F(A 0 ) is supported on the compact submanifold X, and so p 1 (A 0 ) R is defined. Given any other extension A 1 of α, then Chern-Weil theory shows that Tr(F(A 1 ) F(A 1 )) = Tr(F(A 0 ) F(A 0 )) + dγ where γ = 1 0 Tr ( (A 0 A 1 ) 2F(tA 0 + (1 t)a 1 ) ) dt. See, for example, [36, Lemma 3.3]. Since A 0 A 1 vanishes on the collar, Stokes theorem implies that p 1 (A 0 ) = p 1 (A 1 ), so we denote this quantity by p 1 (E, α). Definition 2.3. Given a pair α 0, α 1 of SO(3) connections on the same bundle E over closed 3-manifold Y, define the relative Chern-Simons invariant, cs(y, α 0, α 1 ) R, by cs(y, α 0, α 1 ) = p 1 (R E, α), where α equals α 0 on (, 0] Y and α 1 on [1, ) Y. Definition 2.4. Given a connection α on an SO(3) bundle E over a closed 3- manifold Y, the bundle extend to a bundle with connection A over some 4-manifold Z with boundary Y, since H 3 (BSO(3)) = 0. Define the Chern-Simons invariant of α modulo Z, cs(y, α) R/Z by cs(y, α) = p 1 (A) mod Z. Since the integral of the first Pontryagin form of a connection over a closed 4- manifold is an integer, it follows that cs(y, α) is well defined in R/Z.

5 CHERN-SIMONS INVARIANTS, INSTANTONS, AND Z/2-HOMOLOGY COBORDISM 5 These two invariants are related by cs(y, α 0, α 1 ) = cs(y, α 1 ) cs(y, α 0 ) mod Z whenever the left side is defined. If α extends to a flat connection A over some 4-manifold then p 1 (A) = 0, and hence the Chern-Simons invariant modulo Z is a flat cobordism invariant. In particular, if E Y is the trivial bundle and α a flat connection on E, then cs(y, α) has a canonical lift to R, namely cs(y, θ, α), where θ denotes the trivial connection with respect to the given trivialization of E. Bundle automorphisms h : E E covering the identity which preserve the SO(3) structure group are called gauge transformations. The group G(E) of gauge transformations acts on the space of SO(3) connections A(E) by pullback. If g : E E is a gauge transformation, then cs(y, α, g (α)) equals p 1 (Ẽ), where Ẽ is the bundle over Y S1 obtained by taking the mapping torus of g. Since Y S 1 is closed, Chern-Weil theory implies that p 1 (Ẽ) is an integer, and so we conclude that cs(y, α, g (α)) Z. Notice that if g is homotopic to the identity gauge transformation, then the bundle Ẽ is isomophic to the pullback of E over Y via the projection π : Y S 1 Y, and hence cs(y, α, g (α)) = p 1 (π (E)) = 0. A similar argument shows that the integer cs(y, α, g (α)) depends only on the path component of g in G(E). It follows easily that the reduction modulo Z of cs(y, α 0, α 1 ) depends only on the orbits of α 0 and α 1 under the action of G(E). Similarly, cs(y, α) R/Z depends only on the orbit of α. Dold and Whitney proved that SO(3) bundles over 4-complexes are determined up to isomorphism by their second Stiefel-Whitney class w 2 and their first Pontryagin class p 1 [7]. Moreover, these are related by p 1 P(w 2 ) mod 4, where P : H 2 ( ; Z/2) H 4 ( ; Z/4) denotes the Pontryagin square, a cohomology operation which lifts the cup square H 2 ( ; Z/2) H 4 ( ; Z/2), y y 2. As a consequence, if Y is a Z/2 homology 3-sphere, then H 2 (Y S 1 ; Z/2) = 0, and so cs(y, α, g (α)) 4Z for any gauge transformation g The moduli space. Let (E, α) be an adapted bundle over X. The elliptic operators d αi d αi acting on even degree E Yi valued forms on Y i are selfadjoint with discrete spectrum. Choose (and fix for the remainder of this section) a δ > 0 smaller than the absolute value of any non-zero eigenvalue of any of the d αi d αi. Let f : X [0, ) be a smooth function which equals 0 on X and tδ on {t} Y i for t > 1. Then define the weighted Sobolev spaces L p,δ n of sections of bundles over X as the completion of the space of compactly supported sections with respect to the norm φ L p,δ := ef φ n L p n. Fix an extension A 0 of α to E. Identify E with so(e) via the identification R 3 = so(3) of SO(3) representations, and let SO(E) denote the SO(3) bundle automorphisms. Connections on E take the form A 0 + a for a 1-form a with values in E. Following [9, Chapter 4], define A δ (E, α) = {A 0 + a a L 3,δ 1 (Ω1 X (E))} and G δ (E, α) = {g SO(E) A0 g L 3,δ 1 (T X End(E))} (where A0 denotes the covariant derivative associated to A 0 ). Then pulling back connections extends to an action of the completions G δ (E, α) on A δ (E, α). The

6 6 MATTHEW HEDDEN AND PAUL KIRK moduli space M(E, α) of instantons on (E, α) is defined to be M(E, α) = {A A δ (E, α) F(A) = F(A)}/G δ (E, α). The idea underlying these definitions is that A δ (E, α) consists of connections that limit exponentially along the ends of X to α, and G δ (E, α) consists of gauge transformations that limit exponentially along the ends of X to a gauge transformation that stabilizes α. Definition 2.5. A reducible connection on (E, α) is a connection A in A δ (E, α) whose stabilizer Γ A G δ is non-trivial. Let M(E, α) red M(E, α) denote the subset of gauge equivalence classes of reducible instantons. Its complement M(E, α) \ M(E, α) red is denoted M(E, α). Restricting a gauge transformation to the fiber of E over a base point in X embeds the stabilizer Γ A in SO(3) and identifies it with the centralizer of the holonomy group of A at the base point. In particular, the subgroup G δ 0(E, α) G δ (E, α) of gauge transformations which restrict to the identity in the fiber of E over the base point of X acts freely on A δ (E, α), and the quotient SO(3) = G δ (E, α)/g δ 0(E, α) acts on A δ (E, α)/g δ 0(E, α) with stabilizers Γ A. The restriction of this action to the irreducible instantons defines a principal SO(3) bundle E M(E, α) called the base point fibration [13, Section 9] Calculation of the index. Define Ind + (E, α) to be the Fredholm index of the operator S δ A 0 = d A 0 d + A 0 : L 3,δ 1 (Ω1 X (E)) L 3,δ (Ω 0 X (E) Ω + X (E)) where Ω + denotes the self-dual 2-forms: ω = ω. The Atiyah-Patodi-Singer theorem can be used to calculate this index. Let η(y, α) denote the Atiyah-Patodi-Singer spectral invariant η(b α, 0) of the operator on Y : and let B α : Ω 0 Y (E Y ) Ω 1 Y (E Y ) Ω 0 Y (E Y ) Ω 1 Y (E Y ), B α (φ 0, φ 1 ) = ( d αφ 1, d α φ 0 + d α φ 1 ), ρ(y, α) = η(y, α) 3η(Y ), denote the Atiyah-Patodi-Singer ρ invariant of (Y, α). (In general, for a connection γ on a bundle V Y, ρ(y, γ) = η(y, γ) dim(v ) η(y, θ), where θ is the trivial connection on the trivial 1-dimensional line bundle.) The real number ρ(y, α) depends only on the gauge equivalence class of α. In general, the kernel of B αi equals the direct sum Hα 0 i (Y i ) Hα 1 i (Y i ) of the d αi -harmonic 0- and 1-forms on Y i with coefficients in the flat bundle supporting α i (see [9, Section 2.5.4]). The assumption that each flat connection α i is nondegenerate and the Hodge theorem imply that kerb α = c i=1 Hα 0 i (Y i ). Setting h α = dimhα 0(Y ) and h α i = dimhα 0(Y i) we conclude that h α = h αi. Note that ρ(y i, α i ) = 0 and h αi = 3 when α i has trivial holonomy, as such a connection is gauge equivalent to the trivial connection. The discussion following the proof of Proposition 3.15 of [2] shows that the index Ind + (E, α) is equal to the Atiyah-Patodi-Singer index IndS APS A 0 of the corresponding operator on (the compact manifold) X with Atiyah-Patodi-Singer boundary conditions. This uses the non-degeneracy of the α i to conclude that L 2 solutions

7 CHERN-SIMONS INVARIANTS, INSTANTONS, AND Z/2-HOMOLOGY COBORDISM 7 of S A0 φ = 0 decay along the ends faster than e tδ. Thus Ind + (E, α) can be computed using the Atiyah-Patodi-Singer theorem. For the convenience of the reader we outline the calculation; similar calculations can be found in the literature; for example it is a consequence of (the more general) [28, Proposition 8.4.1]. Let b + (X) denote the dimension of a maximal positive definite subspace of H 2 (X; R) with respect to the intersection form (see Section 2.6 below for the definition of the intersection form in this context). Proposition 2.6. Let (E, α) be an adapted bundle over a path connected 4- manifold X with c 1 boundary components. Assume further that the flat connection α i on each Y i is non-degenerate, and that H 1 (X; Q) = 0 = H 1 (Y ; Q). Then Ind + (E, α) = 2p 1 (E, α) 3(1 + b + (X)) (3 h αi ρ(y i, α i )). α inontrivial Proof. We use the Atiyah-Patodi-Singer index theorem [2] twice, once on the bundle E with operator S APS A 0 = d A 0 d + A 0 : L 2 1(Ω 1 X(E); P + α ) L 2 (Ω 0 X(E) Ω + X (E)) with index IndSA APS 0 = f X,E (R X, F(A 0 )) 1 2 (h α + η(y, α)), X and again on the trivial 1-dimensional R bundle with trivial connection with index S APS = d d + : L 2 1 (Ω1 X ; P + ) L 2 (Ω 0 X Ω+ X ) IndS APS = X f X,ǫ (R X, θ) 1 2 (h + η(y )). In these formulas P α + denotes the spectral projection onto the non-negative eigenspace of B α, and η(y i, α i ) and h α, are as above. The corresponding unadorned symbols refer to their analogues on the trivial, untwisted R bundle ǫ. Since Y i is a rational homology sphere, h = c. The (inhomogeneous) differential form f X,E (R X, F(A)) is a function of the Riemannian curvature R X and the curvature F(A) of a connection A. It equals ( dime 1 8π Tr(F(A) F(A)) )( (χ(r X) + σ(r X )) ), where χ(r X ) is the Euler form and σ(r X ) is the Hirzebruch signature form. Hence f X,E (R X, F(A)) = 2p 1 (A 0 ) + 3 f X,ǫ (R X, θ). X By convention, the integral of an inhomogeneous form over an n-manifold is defined to be the integral of its n-dimensional component. Since Ind + (E, α) is equal to IndSA APS 0, subtracting 3 IndS APS from IndSA APS 0 and simplifying yields (2.2) Ind + (E, α) = 2p 1 (E, α) + 3 IndS APS 1 2 ρ(y, α) (3c h α). To obtain the desired formula, we must show IndS APS = 1 b + (X). The kernel of S APS consists of L 2 harmonic 1-forms on X. Proposition 4.9 of [2] identifies this space with the image H 1 (X, Y ; R) H 1 (X; R). The hypothesis that H 1 (X; Q) = 0 (which holds, in any event, by our standing assumption that H 1 (X; Z/2) = 0) implies that H 1 (X; R) = 0, and so kers APS = 0. X

8 8 MATTHEW HEDDEN AND PAUL KIRK We now identify the cokernel of S APS. The cokernel is isomorphic to the kernel of the adjoint which, in turn, is isomorphic to the extended L 2 solutions to S φ = 0, i.e., the space of pairs φ = (φ 0, φ + ) Ω 0 X Ω + X satisfying dφ 0 + d φ + = 0 and for which (φ 0, φ + ) {t} Y γ decays exponentially on the collar for some harmonic form γ = (γ 0, γ 1 ) kerb on Y [2]. Since Y is a rational homology sphere, kerb = H 0 (Y ). It follows that γ = (γ 0, 0) where γ 0 is a harmonic 0-form, that is, a locally constant function on Y. Thus φ 0 {t} Y γ 0 and φ + {t} Y are exponentially decaying forms. Let X(t) = X ([0, t] Y ). Then dφ 0, d φ + L 2 (X(t)) = ± dφ 0 {t} Y, φ + {t} Y L 2 ({t} Y ) t 0 This implies that dφ 0 and d φ + are L 2 -orthogonal. Therefore, dφ 0 + d φ + = 0 implies that φ 0 is a harmonic 0 form (that is, a constant function) and φ + is an exponentially decaying harmonic self-dual 2-form. Since X is connected we conclude cokers APS = 1 + b + (X), and hence IndS APS = 1 b + (X). If α i has trivial holonomy, then ρ(y, α i ) = 0 and h αi = dimhα 0 i (Y ) = 3. Therefore, (2.3) ρ(y, α) + 3c h α = 3 h αi ρ(y i, α i ) = 3 h αi ρ(y i, α i ) i α inontrivial Combining Equations (2.2) and (2.3) yields the result. Suppose that one of the flat connections, say α 1, on the boundary component Y 1 is trivial with respect to some trivialization of E over Y 1. Suppose further that Z is a compact, negative definite 4 manifold with boundary Y 1 j W j, with the trivialized bundle Z R 3 Z over it. Then glue X to Z along Y 1 to produce the manifold X = X Y1 Z. Let E be the bundle over X obtained by gluing E to the trivial bundle using the given trivializations over Y 1. Extending the connection A 0 over X by the trivial connection over Z gives a new adapted bundle (E, α ) over X. Since p 1(E, α) = p 1 (E, α ), Proposition 2.6 then shows that Ind + (E, α) = Ind + (E, α ). If Ind + (E, α) 0, p 1 (E, α) > 0, and each α i is non-degenerate, then by varying the Riemannian metric inside X if necessary, M(E, α) is a smooth orientable manifold of dimension Ind + (E, α). In the present context this is a consequence of Lemma 8.8.4, Theorem and Remark of [28]. When consulting [28], it is useful to note that because we assume the flat connections α i are non-degenerate, the appropriate path components of the L 2 moduli spaces and thickened moduli spaces of [28] coincide with M(E, α). When p 1 (E, α) = 0, then M(E, α) is the moduli space of flat connections and hence is unchanged by varying the Riemannian metric on X. In this situation one can instead perturb the equation F(A) + F(A) = 0 to make M(E, α) a smooth orientable manifold of dimension Ind + (E, α); see [17, 9] Compactness. A critical question about M(E, α) is whether it is compact. To understand this, we define an invariant ˆτ(Y, α) (0, 4], essentially the minimal relative SO(3) Chern-Simons invariant over the path components of Y. This invariant provides a sufficient condition to guarantee compactness of M(E, α) (Proposition 2.9 below).

9 CHERN-SIMONS INVARIANTS, INSTANTONS, AND Z/2-HOMOLOGY COBORDISM 9 We first introduce some notation. For any path connected space Z and compact Lie group G, denote by χ(z, G) the space χ(z, G) = Hom(π 1 (Z), G)/conjugation. This space is a compact real algebraic variety and is analytically isomorphic to the space of gauge equivalence classes of flat G connections [12]. To a flat G connection α we associate its holonomy hol α χ(z, G). Notice that χ(z, G) is partitioned into disjoint compact subspaces corresponding to the isomorphism classes of G-bundles over Z which support the various flat connections. When Z is not path connected then take χ(z, G) to be the product of the χ(z i, G) over the path components Z i of Z. We can now define the invariant ˆτ(Y, α) (0, 4], which is an adaptation of invariants found in [16, 14] and [28, Definition 6.3.5] to our context. For each boundary component Y i of X, the bundle E Yi carries the flat connection α i and is determined up to isomorphism by the gauge equivalence class of α i. Hence α i determines its second Stiefel-Whitney class. Thus the holonomy gives a decomposition into a disjoint union (2.4) χ(y i, SO(3)) = w H 2 (Y i;z/2) χ(y i, SO(3)) w If γ is any flat connection on E Yi and g any gauge transformation of E Yi, then cs(y i, α i, γ) cs(y i, α i, g (γ)) is an integer, as explained in Section 2.2. This implies that cs(y i, α i, ) descends to a well-defined R/Z valued function on the compact subspace χ(y i, SO(3)) w2(e Yi ) χ(y i, SO(3)). This function is locally constant, see e.g. [21], and hence takes finitely many values in R/Z. This in turn implies that the set S(α i ) = {cs(y i, α i, γ) mod 4 γ a flat connection on E Yi } R/4Z is finite, since it is contained in the preimage of a finite set under R/4Z R/Z. At the moment, taking Chern-Simons invariants mod 4 might seem unmotivated, but it occurs for two reasons. First, in the compactness result below, energy can only bubble off at interior points in multiples of 4. Second, if Y i is a Z/2 homology sphere, then cs(y i, α i, γ) cs(y i, α i, g (γ)) = cs(y i, g (γ), γ) is in fact always four times an integer, as explained in Section 2.2. Definition 2.7. Let b : R/4Z (0, 4] be the obvious bijection. Define τ(y i, α i ) (0, 4] to be the minimal value which b attains on the set S(α i ). Informally, τ(y i, α i ) is the minimal relative Chern-Simons invariant cs(y i, α i, γ) for γ χ(y i, SO(3)) w2(e Yi ), taken modulo 4. Then define (2.5) ˆτ(Y, α) = min Y i X {τ(y i, α i )} (0, 4] Note that ˆτ(Y, α) depends only on Y = i Y i and the gauge equivalence classes of the flat connections α i. The following lemma will prove useful for estimating ˆτ(Y, α). We omit the proof, which is a simple consequence of the remarks in the paragraph following Definition 2.3.

10 10 MATTHEW HEDDEN AND PAUL KIRK Lemma 2.8. Let E Y be an SO(3) vector bundle over a closed 3-manifold and α, γ two flat connections on E. Choose any pair (W α, E α ) where W α is a 4- manifold with boundary Y, E α W α is a vector bundle extending E. Similarly choose (W γ, E γ ). Then cs(y, α, γ) mod 4, taken in (0, 4], is greater than or equal to the fractional part of p 1 (E γ, γ) p 1 (E α, α), taken in (0, 1]. In particular, if p 1 (E γ, γ) is rational with denominator dividing k Z >0 for all flat connections γ on E, then τ(y, α) 1 k for all α. As mentioned, ˆτ(Y, α) provides a sufficient condition for compactness of the moduli space. Proposition 2.9. Suppose that (E, α) is an adapted bundle with each α i nondegenerate and 0 p 1 (E, α) < ˆτ(Y, α). Then M(E, α) is compact. Proof. This follows from the convergence with no loss of energy theorem of Morgan-Mrowka-Ruberman [28, Theorem 6.3.3] which says that a sequence of gauge equivalence classes of finite energy instantons on (E, α) have a geometric limit which is the union of an idealized instanton on X and a union of finite energy instantons on tubes (see also [33] and [15] for earlier versions of this result). We give an outline and refer the reader to [28] for details. This result states that any sequence of gauge equivalence classes of instantons {[A i ]} i=1 in M(E, α) has a subsequence which converges in the following sense. The limit is a sequence of connections (A, B 1,, B k ), where A is an idealized instanton on an adapted bundle (E, α ) over X, and B j, j = 1,, k are instantons on the bundle R E Yij over R Y ij. Here i j {1, 2,, c}. More precisely, A is a smooth instanton away from a finite set of points {x m } in X, and the curvature density F(A ) 2 is the sum of a non-negative smooth function and a Dirac measure supported on these points with weight 32π 2 n m for some non-negative integers n m. The instantons B j on the cylinders have finite, positive energy: 0 < F(B j ) 2 <. In particular none of the B j are flat. The connections A and B j have compatible boundary values, which means that the formal sums of gauge equivalence classes of flat connections on Y = X are equal α α c = α α c + k j=1 ( lim r B j(r) lim r B j(r) ), where B j (r) = B j {r} Yij. Finally, no energy is lost, i.e. there is an equality of non-negative numbers (2.6) p 1 (E, α) = p 1 (E, α ) + k 4n m + p 1 (B j ). {x m} Since p 1 (E, α) < 4 and every term in (2.6) is non-negative, the idealized instanton A is necessarily an (honest) instanton, that is, each n m is zero, the set {x m } is empty, the bundle E equals E, and A is a smooth instanton on E. Informally, no energy can bubble off at interior points. Next, suppose that k > 0. Then the identification of the limiting flat connections implies that one of the instantons B j on R E Yij over R Y ij has left handed limit one of the α i, that is, lim r B j (r) = α ij. Denote by γ the limit j=1

11 CHERN-SIMONS INVARIANTS, INSTANTONS, AND Z/2-HOMOLOGY COBORDISM 11 lim r B j (r). Then p 1 (B j ) cs(y ij, α ij, γ) mod 4, and since B j is not flat, p 1 (B j ) > 0. Therefore p 1 (B j ) τ(y ij, α ij ) ˆτ(Y, α) > p 1 (E, α), which is impossible, since every term in (2.6) is non-negative. Therefore k = 0. Informally, no energy can escape down the ends of X. Thus A M(E, α), as desired Reducible connections. We now discuss reducible adapted bundles and reducible connections. We first remind the reader of the extended intersection form of a 4-manifold whose boundary is a union of rational homology spheres. Start with the pairing H 2 (X; Z) H 2 (X, Y ; Z) Z, x y = (x y) [X, Y ]. Let d be any positive integer so that d H 2 (Y ; Z) = 0. Then this produces a well-defined pairing (2.7) H 2 (X; Z) H 2 (X; Z) 1 d Z by x y := 1 d (x z), where z H2 (X, Y ; Z) satisfies i (z) = d y. This can also be described as the restriction of the composite H 2 (X; Z) H 2 (X; Z) H 2 (X; R) H 2 (X; R) = H 2 (X, Y ; R) H 2 (X, Y ; R) H 4 (X, Y ; R) = R. Note that x y = 0 if x or y is a torsion class. Thus we say X has a negative definite intersection form or more briefly X is negative definite if x x < 0 whenever x is not a torsion class. One defines positive definite analogously. Call X indefinite if it is neither positive nor negative definite. Let b + (X) denote the maximal dimension of any subspace of H 2 (X; R) on which the intersection form is positive definite, and similarly define b (X). The assumption that X has boundary a union of rational homology 3-spheres implies that dim(h 2 (X; R)) = b + (X) + b (X). Therefore b + (X) = 0 if and only if X is negative definite. If X is indefinite, then for a generic Riemannian metric the moduli space M(E, α) is a smooth manifold. However, in contrast to some arguments in gauge theory, the argument of [14, 16] makes use of the fact that reducibles cannot be perturbed away when X is negative definite. This technique originates in Donaldson s proof of his celebrated theorem that definite intersection forms of closed smooth 4-manifolds are diagonalizable [8]. Therefore we make the following assumption for the remainder of this section: The intersection form of X is negative definite; that is, x x < 0 unless x is a torsion class. Near the orbit of a reducible instanton A, an application of the slice theorem shows that M(E, α) has the structure V/Γ A for a vector space V of dimension Ind + (E, α) + dimγ A [9, Theorem 4.13]. This is understood by considering the base point fibration (described above) SO(3) A δ (E, α)/g0 δ (E, α) A δ (E, α)/g δ (E, α). Restricting to M(E, α) gives an SO(3) action on a smooth manifold of dimension Ind + (E, α) + 3 with quotient M(E, α) and stabilizer Γ A over the instanton A.

12 12 MATTHEW HEDDEN AND PAUL KIRK Subgroups of SO(3) have several possible centralizers, and this can lead to different types of singularities in the instanton moduli space. For our purposes, it is sufficient to deal solely with orbit types of reducibles that have SO(2)-stabilizers. In light of our assumption that H 1 (X) has no 2-torsion, we can ensure this with the following (which, hereafter, will be assumed unless otherwise stated): The adapted bundle (E, α) is non-trivial. Equivalently, α does not extend to a connection on E with trivial holonomy. In typical applications either the bundle E is non-trivial, or else one of the α i has non-trivial holonomy, and so the assumption holds. With this assumption, the singularities of M(E, α) are cones on CP n, where 2n + 1 = Ind + (E, α). If n > 0 the second Stiefel-Whitney class of the base point fibration E M(E, α) restricts on the link of each reducible instanton to the generator of H 2 (CP n ; Z/2) = Z/2 [13, Section 9]. Suppose there exists a reducible connection A A δ (E, α). Since Γ A centralizes the holonomy group of A, the assumptions that (E, α) is non-trivial and H 1 (X; Z/2) = 0 imply that Γ A is conjugate to SO(2) SO(3). Moreover, for each boundary component Y i, α i is a reducible flat connection on Y i whose stabilizer contains a maximal torus. The action of Γ A on the fiber E x of E over a point x X gives a splitting E x = L x ǫ x, where ǫ x is the 1-eigenspace of Γ A and L x is its 2-dimensional orthogonal complement on which Γ A acts non-trivially. Parallel transport using A then gives a decomposition E = L A ǫ A into the sum of an orthogonal plane bundle bundle L A and a real line bundle ǫ A. The bundle ǫ A is trivial since H 1 (X; Z/2) = 0. A choice of trivialization of ǫ A together with the orientation of E determines an orientation (and hence the structure of an SO(2) vector bundle) on L A. The connection A splits correspondingly as the direct sum of connections A = a θ, where s an SO(2) connection on L A and θ is the trivial connection on the trivial bundle. In particular, the connection α i on each end splits as α i = β i θ on L A Yi ǫ, where β i = lim r a {r} Yi. The flat connection β i is essentially uniquely determined by the subbundle L A Y E Y, as the following lemma shows. Lemma Every SO(2) vector bundle L over a rational homology sphere Z admits a flat SO(2) connection. Moreover, given any two flat SO(2) connections β, β on L, there exists a path of SO(2) gauge transformations g t with g 0 the identity and g 1 (β ) = β. Proof. For simplicity identify SO(2) with U(1). Choose any U(1) connection b on L. Since H 2 (Z; R) = 0, the real form i 2πF(b) is exact. Adding a 1-form iω to b changes F(b) to F(b)+idω, and so there exists an ω for which F(b + iω) = 0, i.e. β = b + iω is flat. The group of gauge transformations of the bunlde L, G(L) = C (Z, U(1)), has a single path component, since [Z, U(1)] = H 1 (Z; Z) = 0. A flat connection β on L has a holonomy representation hol β χ(π 1 Z, U(1)) = Hom(π 1 Z, U(1)) = H 1 (Z; R/Z). The Bockstein in the coefficient exact sequence H 1 (Z; R/Z) H 2 (Z; Z) takes hol β to c 1 (L) (see below) and is an isomorphism since Z is a rational homology sphere.

13 CHERN-SIMONS INVARIANTS, INSTANTONS, AND Z/2-HOMOLOGY COBORDISM 13 Since the holonomy determines the gauge equivalence class of a flat connection, it follows that any two flat connections β, β on L are gauge equivalent, and that there is a path g t of gauge transformations with g 0 = Id and g1 (β ) = β. The relationship between the holonomy U(1) representation and the first Chern class used in the proof of Lemma 2.10 can be explained in the following way. Let U(1) d denote U(1) with the discrete topology. The identity map i : U(1) d U(1) induces a map Bi : BU(1) d BU(1) on classifying spaces. Since BU(1) d = K(R/Z, 1) and BU(1) = K(Z, 2), we have a sequence of (natural) isomorphisms for any finite abelian group A Hom(A, U(1)) = H 1 (K(A, 1); R/Z) = [K(A, 1), BU(1) d ] Bi [K(A, 1), BU(1)] = H 2 (K(A, 1); Z). The identity map BU(1) BU(1), viewed as a class in H 2 (BU(1); Z), is the first Chern class of the universal line bundle, and so naturality implies that a homomorphism h : A U(1) is sent to c 1 (L), where L is the pull back of the universal bundle via K(A, 1) Bh K(U(1) d, 1) Bi BU(1). Moreover, the map Bi can be identified with the Bockstein. Given a manifold W equipped with a homomorphism h : π 1 (W) U(1) with finite image, view h as an element of H 1 (W; R/Z). Then h is sent by the Bockstein to the first Chern class of the U(1) bundle that supports a flat connection with holonomy h. In the case of a rational homology 3-sphere Z, we have a sequence of isomorphisms Hom(π 1 (Z), Q/Z) = Hom(H 1 (Z), Q/Z) = H 1 (Z; Q/Z) B H 2 (Z; Z) [Z] H 1 (Z; Z) with B the Bockstein and the last isomorphism Poincaré duality. The inverse of this composite is given by the linking form, m lk(m, ) : H 1 (Z; Z) Q/Z. Thus we have shown the following. Lemma Suppose Z is a rational homology 3-sphere and c 1 H 2 (Z; Z) a class with Poincaré dual m H 1 (Z; Z). Then the holonomy of the flat U(1) bundle with first Chern class c 1 is given by hol(γ) = lk(m, γ), where is the linking form. lk : H 1 (Z; Z) H 1 (Z; Z) Q/Z exp(2πi ) U(1) Identifying U(1) and SO(2) via C = R 2 allows us to make the analogous statement, substituting the Euler class for c 1. Suppose that (E, α) is an adapted bundle which admits a reducible connection A. Then (E, α) = (L ǫ, β θ), where β is unique in the sense of Lemma Conversely, if ǫ E is a trivializable real line bundle and L the orthogonal 2-plane bundle, then orient ǫ (and hence L). Let β denote the flat SO(2) connection carried by L over the end and set α = β θ. Extend β to a connection a on L. Then A = a θ is a reducible connection in A(E, α). This establishes the first assertion in the following lemma. Lemma An adapted bundle (E, α) supports a reducible connection if and only if there exists a nowhere zero section s of E, and hence a splitting E = L ǫ

14 14 MATTHEW HEDDEN AND PAUL KIRK with L the orthogonal complement of ǫ = spans, and, on the ends, α = β θ where β is the (unique up to homotopy) flat connection on L. If an adapted bundle (E, α) supports a reducible connection inducing a splitting E = L ǫ, it supports a reducible instanton in M(E, α) inducing the same splitting. Proof. The first part was justified in the preceding paragraph. Suppose that (E, α) = (L, β θ). Since Y is a union of rational homology spheres, Proposition 4.9 of [2] shows that the space of L 2 harmonic 2-forms {ω L 2 (Ω 2 X ) dω = 0 = d ω} maps isomorphically to H 2 (X; R). Thus there exists an L 2 harmonic 2-form ω on X representing the class e(l) H 2 (X; R). In light of the facts that X is negative definite, the Hodge -operator preserves L 2 harmonic 2-forms, and 2 = 1, it follows that ω = ω. Choose a connection a on L. Then 1 2π Pf(F(a)) and ω are cohomologous by Chern-Weil Theory, where Pf denotes the pfaffian of an so(2) matrix. By adding an so(2) valued 1-form to a one can change Pf(F(a)) arbitrarily within its cohomology class. Thus we may choose a so that 1 2π Pf(F(a)) = ω. The connection a limits to a flat connection on L along the ends. Thus Lemma 2.10 implies a limits to h 1 (β) for some gauge transformation h : L Y L Y which can be connected to the identity by a path of gauge transformations. Extend h to a gauge transformation h of all of L by the identity on X, the path from the identity to h on [0, 1] Y, and h on [1, ) Y. Replacing a by h(a) does not change F(a) since SO(2) is abelian, and h(a) limits to β. Let A = h(a) θ on the SO(3) vector bundle L ǫ. Then A is a reducible instanton on (E, α) = (L ǫ, β θ) with L 2 -curvature. Since we assumed that the flat connection α on Y is non-degenerate, it follows from [9, Theorem 4.2] that A L p,δ 1 and so A M(E, α). The following proposition determines the Pontryagin charge of a reducible instanton in terms of the intersection form. Proposition Let e H 2 (X; Z). Let L X be the SO(2) vector bundle with Euler class e. Let β be the flat connection on the restriction L Y, and (E, α) = (L ǫ, β θ) the corresponding adapted bundle. Then p 1 (E, α) = e e 1 d Z and w 2(E) e mod 2. Proof. Choose (arbitrarily) a connection a 0 on L which extends the flat connection β on the ends, and let A 0 = a 0 θ be the corresponding connection on E = L ǫ; this connection agrees with α = β θ over the ends, and hence lies in A δ (E, α). Let ê denote the differential 2-form on X ê = 1 2π Pf(F(a 0)). A straightforward calculation starting with the formula for the inclusion of Lie algebras so(2) so(3) gives 1 8π 2 Tr(F(A 0) F(A 0 )) = 1 2 Tr( 1 2π F(A 0) 1 2π F(A 0)) = 1 2π Pf(F(a 0)) 1 2π Pf(F(a 0)) = ê ê.

15 CHERN-SIMONS INVARIANTS, INSTANTONS, AND Z/2-HOMOLOGY COBORDISM 15 Hence p 1 (A 0 ) = p 1 (E, α) = ê ê. X Chern-Weil theory implies that ê is a closed 2-form. It vanishes on the ends since a 0 is flat on the ends. On a closed 4-manifold Z Chern-Weil theory implies that ê represents (in DeRham cohomology) the image of the Euler class e under the coefficient homomorphism H 2 (Z; Z) H 2 (Z; R). Since H 2 (X; R) = Hom(H 2 (X; Z), R) and every class in H 2 (X; Z) is represented by a closed surface, it follows that ê represents the image of e in H 2 (X; R). For x, y H 2 (X, Y ; R) represented by closed forms that vanish on the boundary, x y = X x y. Since H2 (X, Y ; R) H 2 (X; R) is an isomorphism, it follows that p 1 (E, α) = ê ê = e e. X The congruence w 2 (E) e mod 2 follows from the general congruence w 2 (L ǫ) e(l) mod 2, which is valid for any SO(2) bundle L over any space Enumeration of reducible instantons. We next address the problem of enumerating the reducible instantons in M(E, α). The results of the previous section allow us to reduce this problem to the enumeration of bundle reductions. These in turn determine certain cohomology classes and motivate the following definition. Definition Let (E, α) = (L ǫ, β θ) with L non-trivial. Fix an orientation of ǫ, orienting L. Let e = e(l) H 2 (X; Z) denote the Euler class of L. Define C(e) = {e H 2 (X; Z) e e = e e, e e mod 2, i j (e ) = ±i j (e), j = 1,, c}/±1 where i j : Y j Y X denotes the inclusion of the jth boundary component. Our standing assumption that X is negative definite implies that C(e) is a finite set. Suppose that L, L are two orthogonal plane subbundles of E and that (E, α) = (L ǫ, β θ) = (L ǫ, β θ). Orient L and L arbitrarily and let e = e(l), e = e(l ). Then e e mod 2 and e e = e e by Proposition Fix a boundary component Y j of X. If α j is non-trivial, then since β j θ = α j = β j θ, the unoriented subbundles L Yj and L Yj are equal, since their orthogonal complements coincide. But if α j is the trivial connection, L Yi and L Yi need not coincide, although they are both trivial bundles. In either case, i j (e) = ±i j (e ). Changing the orientation of L changes the sign of e. Since X is connected, it follows that the function taking a reducible connection A A δ (L ǫ, β θ) red to e(l A ) C(e) is well-defined. Proposition The function Φ : M(E, α) red C(e) defined by Φ([A]) = e(l A )/±1 is well-defined and injective. If C(e) consists of a single point then Φ is a bijection. Proof. We first show that Φ is well-defined on gauge equivalence classes. Suppose that A is a reducible instanton on (E, α) and g G δ (E, α) a gauge transformation. Then the decomposition E = L A ǫ is sent to E = g(l A ) g(ǫ). Hence L A and g(l A ) are isomorphic bundles over X, and so Φ(A) = Φ(g(A)). Thus Φ is well-defined on gauge equivalence classes.

16 16 MATTHEW HEDDEN AND PAUL KIRK Suppose that A and B are reducible instantons on (E, α) which satisfy Φ(A) = Φ(B). Then, after reorienting L B and ǫ B if needed, e(l A ) = e(l B ). Since SO(2) bundles are determined up to isomorphism by their Euler class, there exists a bundle isomorphism h : L A L B. Extending by the unique orthogonal orientationpreserving isomorphism I : ǫ A ǫ B produces a bundle isomorphism g = h I : E = L A ǫ A L B ǫ B = E. Since the decompositions agree on the ends with the given decomposition (E, α), lim r g {r} Yj exists and lies in the stabilizer of α j, so that g G δ (E, α). Therefore, A and B represent the same point in M(E, α). Lemma 2.12 shows that M(L ǫ, β θ) is non-empty, and so if C(e) = {e}, Φ is surjective. Determining the image of Φ can be tricky when Y is not a union of Z/2 homology spheres. Lemma 2.12 shows that e = e(l) itself is always in the image, and hence, as noted in Proposition 2.15, if C(e) consists of a single point, then Φ is a bijection. Theorem Let (E, α) = (L ǫ, β θ) be a reducible adapted bundle with Euler class e = e(l) and let Φ : M(E, α) red C(e) be the injective function defined above. (1) If H 2 (X; Z) splits orthogonally with respect to the intersection form as H 2 (X; Z) = span(e) W with span(e) = Z and W free abelian, then C(e) consists of a single point and so Φ is a bijection. (2) To each e C(e) one can assign an obstruction in H 1 (Y ; Z/2) whose vanishing implies that e = Φ([A]) has a solution [A] M(E, α) red. (3) If each component of Y is a Z/2 homology sphere, then Φ is a bijection. Proof. Suppose first that H 2 (X; Z) splits orthogonally with respect to the intersection form as H 2 (X; Z) = span(e) W. Then any class e satisfying e e mod 2 takes the form e = (1 + 2k)e + 2w, with w W. Hence e e = (1 + 4k + 4k 2 )e e + 4w w. If e C(e) then e e = e e, so that 4(k 2 + k)e e + 4w w = 0. Since e e < 0 and w w 0, this is possible if and only if k = 0 or 1 and w = 0, so that e = ±e, as claimed. We turn now to the second assertion, which clearly implies the third assertion. Given a class e C(e), let L X be an SO(2) vector bundle with e = e(l ). Then since e e mod 2, the bundles E = L ǫ and L ǫ over X are isomorphic. Fix an identification E = L ǫ = L ǫ. Let β j be the unique flat connection on L Yj, β j be the unique flat connection on L Yj. Then α j = β j θ and α j = β j θ are two flat connections on E Yj. If there exists a gauge transformation g : E E so that g Y j (α j ) = α j, then Lemma 2.12 implies that there exists an instanton A M(E, α) inducing the splitting E = g(l ) g(ǫ ), and hence e = Φ(A ) has the solution A. So we seek to construct such a gauge transformation g. We construct g first on the boundary and the try to extend over the interior. For each j, choose an isomorphism h j : L Yj L Yj which preserves or reverses orientation according to whether i j (e ) = i j (e) or i j (e ) = i j (e) (if 2i j (e) = 0 choose h j to preserve orientation, for definiteness). Then h j (β j ) and β j are flat connections on L Yj and so we may assume, by modifying β j if needed, that h j (β j ) = β j. Let g j : E Yj = L Yj ǫ E = L Yj ǫ be the unique orientationpreserving orthogonal extension (so that on the trivial R factors, the orientation is

17 CHERN-SIMONS INVARIANTS, INSTANTONS, AND Z/2-HOMOLOGY COBORDISM 17 preserved or reversed according to the sign). By construction, gj (α j ) = α j. Let g = j g j : E Y E Y. The stabilizer of α j is connected except for those j for which i j (e) has order 2, in which case the stabilizer is isomorphic to O(2). Thus the desired gauge transformation exists if and only if g can be extended to a gauge transformation over X, after perhaps composing g j with an appropriate gauge transformation k j for those path components such that i j (e) has order 2. Let P be the principal SO(3) bundle associated to E X, so that gauge transformations are sections of P Ad SO(3). Obstruction theory identifies the primary obstruction to extending g to all of X as a class ψ H 2 (X, Y ; π 1 (SO(3))) = H 2 (X, Y ; Z/2). Naturality of primary obstructions, together with the fact that P Ad SO(3) admits sections (e.g. the identity transformation) shows that ψ is in kerh 2 (X, Y ; Z/2) H 2 (X; Z/2), hence ψ lies in the image of the injection H 1 (Y ; Z/2) H 2 (X, Y ; Z/2). Let ϑ H 2 (Y ; Z/2) denote the unique element mapped to ψ. When ϑ = 0, then g extends over the 2-skeleton of X. It further extends further to the 3-skeleton, because π 2 (SO(3)) = 0. The last obstruction to extending g over X lies in H 4 (X, Y ; π 3 (SO(3))) = H 4 (X, Y ; Z); the coefficients are untwisted since our standing assumption that H 1 (X; Z/2) = 0 implies that π 1 (X) admits no nontrivial homomorphisms to Aut(Z) = Z/2. This class is the obstruction to extending g over the top cell of X/Y. Suppose that D X is a fixed 4-ball, and that g : E X\D E X\D is the gauge transformation which we wish to extend to all of E. Choose a connection A A δ (L ǫ, β L θ). The connection g (A X\D ) extends to a connection B A δ (L ǫ, β L θ ). Since Tr(F(g (A)) F(g (A))) = Tr(F(A) F(A)) pointwise (on X \ D), it follows that (2.8) p 1 (A) p 1 (B) = 1 8π 2 D Tr(F(A) F(A)) 1 8π 2 D Tr(F(B) F(B)), The condition e e = e e implies that p 1 (A) = p 1 (B), and hence the left side of Equation (2.8) is zero. But Chern-Weil theory implies that the right side of Equation (2.8) equals p 1 (V ), where V is the bundle over S 4 obtained by clutching two copies of E D using g D. Since p 1 (V ) = 0, the bundle V is trivial, and thus the restriction of g to D extends over D, giving the desired gauge transformation g. Thus we have seen that there is a class ϑ H 1 (Y ; Z/2) whose vanishing guarantees that g extends over X The argument of Furuta. With all the necessary machinery in place, we can now describe our variant of the argument of Furuta. Theorem 2.17 provides the mechanism to establish that certain definite 4-manifolds cannot exist. Furuta used this argument to prove the linear independence of the Seifert fibered homology spheres {Σ(p, q, pqk 1)} k=1 in Θ3 H. We will give further applications below. Recall from Section 2.6 that if X is a negative definite four-manifold satisfying H 1 (X; Z/2) = 0, and X consists of a union of rational homology spheres, then any SO(2) bundle L determines a unique adapted SO(3) bundle (E, α) = (L ǫ, β θ). Letting e(l) denote the Euler class of L, Proposition 2.13 shows that p 1 (E, α) =

18 18 MATTHEW HEDDEN AND PAUL KIRK e(l) e(l) 1 d Z, where d is an integer which annihilates H 1(Y i ; Z) for each i. Finally, recall the invariant ˆτ(Y, α) (0, 4] of Definition 2.7. Theorem Let X be a negative definite 4-manifold with H 1 (X; Z/2) = 0 whose boundary consists of a union of rational homology spheres. Let L be an SO(2) bundle over X with Euler class e = e(l), and let (E, α) = (L ǫ, β θ) be the corresponding adapted bundle. Assume that (1) Each α i = β i θ is non-degenerate, (2) Ind + (E, α) 0, where Ind + (E, α) = 2e e (3) 0 < e e < ˆτ(Y, α) 4. β inontrivial (3 h αi ρ(y i, α i )). Then the number of gauge equivalence classes of reducible instantons on (E, α) is even, greater than 1, and Ind + (E, α) is odd. Proof. The irreducible moduli space M(E, α) is a smooth oriented manifold of dimension Ind + (E, α). Let k denote the number of singular points in the full moduli space. Proposition 2.15 identifies k with the image of Φ : M(E, α) red C(L), and shows that k 1. Each singular point A M(E, α) red has a neighborhood homeomorphic to a cone on CP n for some n, and so 2n + 1 = Ind + (E, α). Proposition 2.9 implies that M(E, α) is compact. Removing cone neighborhoods of the k singular points yields a 2n + 1 dimensional compact manifold M 0, whose boundary consists of a disjoint union of k copies of CP n (perhaps with differing orientations). This manifold is equipped with an SO(3) bundle E M 0 that restricts to a bundle on each CP n with nontrivial second Stiefel-Whitney class. Since H (CP n ; Z/2) is a polynomial ring on w 2, 0 w 2 (E CP n) n for each boundary component. But since ( k i=1 CP n, E) extends to (M 0, E), it follows that k is even, as this is the only way for the Stiefel-Whitney numbers of ( k i=1 CP n, E) to vanish Computing Atiyah-Patodi-Singer ρ invariants and Chern-Simons invariants of flat SO(2) connections on rational homology spheres. Before we turn to applications and examples we discuss the use of flat cobordisms to compute the ρ invariants and Chern-Simons invariants which appear in the expression for Ind + (E, α) and the definition of ˆτ(Y, α). The Atiyah-Patodi-Singer ρ invariants of flat SO(3) connections on 3-manifolds can be difficult to compute. However, in Theorem 2.17 we require Y to be a union of rational homology spheres and the flat connection α to be reducible; in fact, to take the form α = β θ for θ the trivial connection. Thus, what is required in applications of this theorem is a calculation of ρ invariants of such reducible flat SO(3) connections on rational homology spheres. This is an easier task, and can often be accomplished as follows. Since ρ(y, α) depends only on the gauge equivalence class of α, and taking the holonomy induces a 1-1 correspondence from gauge equivalence classes of flat connections to conjugacy classes of homomorphisms π 1 (Y ) SO(3), we henceforth use the notation α for both a flat connection and its holonomy.